Question

If $\Delta$ ABC and $\Delta$ DEF are similar such that $2 \text{ AB} = \text{DE}$ and $\text{BC} = 8 \text{ cm}$, then EF is equal to :

• A. 4 cm
• B. 8 cm
• C. 12 cm
• D. 16 cm

Hint:

The relation $2 \text{ AB} = \text{DE}$ tells us that each side of the second triangle ($\Delta \text{DEF}$) is exactly twice the length of the corresponding side of the first triangle ($\Delta \text{ABC}$).
Since $\text{BC} = 8 \text{ cm}$, EF must be $2 \times 8 = 16 \text{ cm}$.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given two similar triangles, $\Delta \text{ABC}$ and $\Delta \text{DEF}$.
We are given a relationship between the corresponding sides $2 \text{ AB} = \text{DE}$ and the length of one side $\text{BC} = 8 \text{ cm}$.
We need to find the length of the corresponding side EF in the second triangle.

Step 2: Key Formula or Approach:
If two triangles are similar, then their corresponding sides are in the same ratio.
Given $\Delta \text{ABC} \sim \Delta \text{DEF}$, we can write the ratio of corresponding sides as:
\[ \frac{\text{AB}}{\text{DE}} = \frac{\text{BC}}{\text{EF}} = \frac{\text{AC}}{\text{DF}} \]

Step 3: Detailed Explanation:

• Write down the given relation between sides AB and DE:
\[ 2 \text{ AB} = \text{DE} \]
Rearranging this equation to find the ratio $\frac{\text{AB}}{\text{DE}}$:
\[ \frac{\text{AB}}{\text{DE}} = \frac{1}{2} \]

• Since the triangles are similar ($\Delta \text{ABC} \sim \Delta \text{DEF}$), the ratio of the corresponding sides BC and EF must equal the ratio of AB and DE:
\[ \frac{\text{AB}}{\text{DE}} = \frac{\text{BC}}{\text{EF}} \]

• Substitute the known values into this proportion:
\[ \frac{1}{2} = \frac{8}{\text{EF}} \]

• Solve for EF using cross-multiplication:
\[ 1 \times \text{EF} = 8 \times 2 \]
\[ \text{EF} = 16 \text{ cm} \]

• Thus, the length of the corresponding side EF is $16 \text{ cm}$.


Step 4: Final Answer:
The length of EF is 16 cm.
Hence, option (D) is correct.